Efficient solution of full domain 3D electromagnetic modelling problems

Abstract

Electromagnetic modelling of 3D conductive targets using staggered grids results in matrix systems that are severely ill conditioned and unsuitable for many iterative solvers. Either a minimal residual solver, MINRES, or a restarted biconjugate gradient stabilised solver, BiCGSTAB(l), can be used to bring the solution to convergence. The BiCGSTAB(l), method, combining the bi-conjugate gradient method with a restarted minimal residual correction, is very efficient in solving the matrix system arising in our applications. To speed up the solver further we employ various preconditioning techniques. The use of a symmetric Jacobian preconditioner greatly improves the convergence of the MINRES method. In homogeneous regions where the divergence of the electric field is zero the Maxwell equations reduce to vector Helmholtz equations with decoupled components. These Helmholtz equations can be readily approximated by central differences, resulting in diagonally dominant matrices that can be used as preconditioners for the original curl curl equation of the staggered grid. The diagonals of the Helmholtz equation can also be used as a Jacobian preconditioner, which is superior to Jacobian preconditioning using the diagonal elements of the original curl curl equation. Numerical results have demonstrated the efficiency of these approaches.